This was the case for me, though I wouldn't classify as being "screwed."
I attended a private tech school, and they were upfront they would not accept credits for any engineering program pre-reqs (Math, Chem, CS, Tech Comm, etc). However, they would accept "humanities" credits an apply them to any electives required for our chosen degree.
Luckily, I had participated in our schools "running start" so I earned those credits while in high school, and my only intention of taking math at the time was to fulfill my highschool math requirements. I did also take 2-years of mandarin, which my school gladly accepted and counted towards my electives.
All that being said, in Washington state, all publicly funded schools must accept all credits from Community Colleges, AND guarantee admission, so if you are a student looking to attend a state school in Washington state, community college is a very attractive route.
For the truly frugal student, I would probably recommend something like what I did: take community college or AP classes aplenty in your senior year. Go straight to a college that has the best program for your interests (keeping in mind your interests may change). Graduating in three years is easy enough if you have a semester’s worth of transfer credits for gen eds, and classes like calculus and linear algebra in particular are really easy to cover before you go to college. Administration will probably try to make graduating early as hard as possible, but they really won’t be able to stop you if you have the credits already.